Question

conceptual question: a marble is placed on the rim of a bowl and launched, rolling into the center of the bowl and climbing up to the opposite rim, turning around and repeating this process over and over again.
a) what type of motion is this, express it mathematically.
b) draw this motion for a given bowl height as a path in the complex plane, using what you know about complex numbers in the polar and Euler format. assume the marble is frictionless
c) how would the path in the complex plane change if the marble experienced friction?

Answer 1

a) It is a simple harmonic motion .

Here, md²x/dt² = - kx

b) Here, x(t) = xocos(wt) + (vo/w)sin(wt)

c)     The path in the complex plane change if the marble experienced friction =>

The simple harmonic motion oscillations are damped .

=>   x(t) = e^-kx * [xocos(wt) + (vo/w)sin(wt)]